LMFDB - Embedded newform 575.3.c.a.574.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [575,3,Mod(574,575)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(575, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("575.574");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 575 = 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	575.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$15.6676152007$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.24681024.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 12x^{4} + 36x^{2} + 9 $ x^6 + 12x^4 + 36x^2 + 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 23)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			574.4
Root			$2.14510i$ of defining polynomial
Character	$\chi$	$=$	575.574
Dual form			575.3.c.a.574.3

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.601466i q^{2} -1.54364i q^{3} +3.63824 q^{4} +0.928445 q^{6} +4.59414i q^{8} +6.61718 q^{9} +O(q^{10})$	$q+0.601466i q^{2} -1.54364i q^{3} +3.63824 q^{4} +0.928445 q^{6} +4.59414i q^{8} +6.61718 q^{9} -5.61612i q^{12} -23.5162i q^{13} +11.7897 q^{16} +3.98001i q^{18} +23.0000i q^{23} +7.09168 q^{24} +14.1442 q^{26} -24.1073i q^{27} +42.4015 q^{29} -27.9663 q^{31} +25.4677i q^{32} +24.0749 q^{36} -36.3005 q^{39} +74.9986 q^{41} -13.8337 q^{46} -93.8839i q^{47} -18.1991i q^{48} -49.0000 q^{49} -85.5575i q^{52} +14.4997 q^{54} +25.5030i q^{58} -26.0000 q^{59} -16.8208i q^{62} +31.8410 q^{64} +35.5037 q^{69} +140.916 q^{71} +30.4003i q^{72} +56.8366i q^{73} -21.8335i q^{78} +22.3418 q^{81} +45.1091i q^{82} -65.4525i q^{87} +83.6795i q^{92} +43.1698i q^{93} +56.4679 q^{94} +39.3128 q^{96} -29.4718i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 24 q^{4} - 66 q^{6} - 54 q^{9}+O(q^{10}) $ 6 * q - 24 * q^4 - 66 * q^6 - 54 * q^9	$ 6 q - 24 q^{4} - 66 q^{6} - 54 q^{9} + 96 q^{16} + 264 q^{24} + 174 q^{26} + 510 q^{36} + 84 q^{39} - 294 q^{49} + 594 q^{54} - 156 q^{59} + 90 q^{64} + 486 q^{81} + 114 q^{94} - 42 q^{96}+O(q^{100}) $ 6 * q - 24 * q^4 - 66 * q^6 - 54 * q^9 + 96 * q^16 + 264 * q^24 + 174 * q^26 + 510 * q^36 + 84 * q^39 - 294 * q^49 + 594 * q^54 - 156 * q^59 + 90 * q^64 + 486 * q^81 + 114 * q^94 - 42 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/575\mathbb{Z}\right)^\times$.

$n$	$51$	$277$
$\chi(n)$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.601466i	0.300733i	0.988630	+	0.150366i	$0.0480453\pi$
$2$	0.601466i	0.300733i	−0.988630	+	0.150366i	$0.951955\pi$
$3$	− 1.54364i	− 0.514546i	−0.966339	−	0.257273i	$-0.917176\pi$
$3$	− 1.54364i	− 0.514546i	0.966339	−	0.257273i	$-0.0828239\pi$
$4$	3.63824	0.909560
$5$	0	0
$5$	0	0
$6$	0.928445	0.154741
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	4.59414i	0.574267i
$9$	6.61718	0.735243
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	− 5.61612i	− 0.468010i
$13$	− 23.5162i	− 1.80894i	−0.426540	−	0.904469i	$-0.640268\pi$
$13$	− 23.5162i	− 1.80894i	0.426540	−	0.904469i	$-0.359732\pi$
$14$	0	0
$15$	0	0
$16$	11.7897	0.736859
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	3.98001i	0.221112i
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	23.0000i	1.00000i
$23$	23.0000i	1.00000i
$24$	7.09168	0.295487
$25$	0	0
$26$	14.1442	0.544007
$27$	− 24.1073i	− 0.892862i
$28$	0	0
$29$	42.4015	1.46212	0.731060	−	0.682314i	$-0.239026\pi$
$29$	42.4015	1.46212	0.731060	+	0.682314i	$0.239026\pi$
$30$	0	0
$31$	−27.9663	−0.902138	−0.451069	−	0.892489i	$-0.648957\pi$
$31$	−27.9663	−0.902138	−0.451069	+	0.892489i	$0.648957\pi$
$32$	25.4677i	0.795865i
$33$	0	0
$34$	0	0
$35$	0	0
$36$	24.0749	0.668747
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	−36.3005	−0.930781
$40$	0	0
$41$	74.9986	1.82924	0.914618	−	0.404320i	$-0.132492\pi$
$41$	74.9986	1.82924	0.914618	+	0.404320i	$0.132492\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	−13.8337	−0.300733
$47$	− 93.8839i	− 1.99753i	−0.0496817	−	0.998765i	$-0.515821\pi$
$47$	− 93.8839i	− 1.99753i	0.0496817	−	0.998765i	$-0.484179\pi$
$48$	− 18.1991i	− 0.379148i
$49$	−49.0000	−1.00000
$50$	0	0
$51$	0	0
$52$	− 85.5575i	− 1.64534i
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	14.4997	0.268513
$55$	0	0
$56$	0	0
$57$	0	0
$58$	25.5030i	0.439707i
$59$	−26.0000	−0.440678	−0.220339	−	0.975423i	$-0.570716\pi$
$59$	−26.0000	−0.440678	−0.220339	+	0.975423i	$0.570716\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	− 16.8208i	− 0.271303i
$63$	0	0
$64$	31.8410	0.497516
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	35.5037	0.514546
$70$	0	0
$71$	140.916	1.98474	0.992368	−	0.123310i	$-0.0393509\pi$
$71$	140.916	1.98474	0.992368	+	0.123310i	$0.0393509\pi$
$72$	30.4003i	0.422226i
$73$	56.8366i	0.778584i	0.921114	+	0.389292i	$0.127280\pi$
$73$	56.8366i	0.778584i	−0.921114	+	0.389292i	$0.872720\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	− 21.8335i	− 0.279916i
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	22.3418	0.275825
$82$	45.1091i	0.550111i
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 65.4525i	− 0.752327i
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	83.6795i	0.909560i
$93$	43.1698i	0.464191i
$94$	56.4679	0.600723
$95$	0	0
$96$	39.3128	0.409509
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	− 29.4718i	− 0.300733i
$99$	0	0
$100$	0	0
$101$	−166.000	−1.64356	−0.821782	−	0.569802i	$-0.807020\pi$
$101$	−166.000	−1.64356	−0.821782	+	0.569802i	$0.807020\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	108.037	1.03881
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	− 87.7080i	− 0.812111i
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	154.267	1.32988
$117$	− 155.611i	− 1.33001i
$118$	− 15.6381i	− 0.132526i
$119$	0	0
$120$	0	0
$121$	121.000	1.00000
$122$	0	0
$123$	− 115.771i	− 0.941225i
$124$	−101.748	−0.820549
$125$	0	0
$126$	0	0
$127$	243.881i	1.92032i	0.279443	+	0.960162i	$0.409850\pi$
$127$	243.881i	1.92032i	−0.279443	+	0.960162i	$0.590150\pi$
$128$	121.022i	0.945484i
$129$	0	0
$130$	0	0
$131$	−182.414	−1.39247	−0.696235	−	0.717814i	$-0.745143\pi$
$131$	−182.414	−1.39247	−0.696235	+	0.717814i	$0.745143\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	21.3542i	0.154741i
$139$	−118.304	−0.851109	−0.425555	−	0.904933i	$-0.639921\pi$
$139$	−118.304	−0.851109	−0.425555	+	0.904933i	$0.639921\pi$
$140$	0	0
$141$	−144.923	−1.02782
$142$	84.7563i	0.596875i
$143$	0	0
$144$	78.0149	0.541770
$145$	0	0
$146$	−34.1853	−0.234146
$147$	75.6382i	0.514546i
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	−188.672	−1.24948	−0.624741	−	0.780832i	$-0.714796\pi$
$151$	−188.672	−1.24948	−0.624741	+	0.780832i	$0.714796\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	−132.070	−0.846601
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	13.4378i	0.0829495i
$163$	314.249i	1.92791i	0.266070	+	0.963954i	$0.414275\pi$
$163$	314.249i	1.92791i	−0.266070	+	0.963954i	$0.585725\pi$
$164$	272.863	1.66380
$165$	0	0
$166$	0	0
$167$	242.000i	1.44910i	0.689221	+	0.724551i	$0.257953\pi$
$167$	242.000i	1.44910i	−0.689221	+	0.724551i	$0.742047\pi$
$168$	0	0
$169$	−384.011	−2.27225
$170$	0	0
$171$	0	0
$172$	0	0
$173$	22.0000i	0.127168i	0.997977	+	0.0635838i	$0.0202530\pi$
$173$	22.0000i	0.127168i	−0.997977	+	0.0635838i	$0.979747\pi$
$174$	39.3674	0.226249
$175$	0	0
$176$	0	0
$177$	40.1346i	0.226749i
$178$	0	0
$179$	−287.187	−1.60440	−0.802198	−	0.597059i	$-0.796336\pi$
$179$	−287.187	−1.60440	−0.802198	+	0.597059i	$0.796336\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	−105.665	−0.574267
$185$	0	0
$186$	−25.9651	−0.139598
$187$	0	0
$188$	− 341.572i	− 1.81687i
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	− 49.1510i	− 0.255995i
$193$	36.1432i	0.187271i	0.995607	+	0.0936353i	$0.0298488\pi$
$193$	36.1432i	0.187271i	−0.995607	+	0.0936353i	$0.970151\pi$
$194$	0	0
$195$	0	0
$196$	−178.274	−0.909560
$197$	− 219.461i	− 1.11402i	−0.830507	−	0.557008i	$-0.811949\pi$
$197$	− 219.461i	− 1.11402i	0.830507	−	0.557008i	$-0.188051\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	0	0
$202$	− 99.8433i	− 0.494274i
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	152.195i	0.735243i
$208$	− 277.250i	− 1.33293i
$209$	0	0
$210$	0	0
$211$	−406.000	−1.92417	−0.962085	−	0.272749i	$-0.912067\pi$
$211$	−406.000	−1.92417	−0.962085	+	0.272749i	$0.912067\pi$
$212$	0	0
$213$	− 217.524i	− 1.02124i
$214$	0	0
$215$	0	0
$216$	110.752	0.512741
$217$	0	0
$218$	0	0
$219$	87.7351	0.400617
$220$	0	0
$221$	0	0
$222$	0	0
$223$	382.000i	1.71300i	0.516143	+	0.856502i	$0.327367\pi$
$223$	382.000i	1.71300i	−0.516143	+	0.856502i	$0.672633\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	194.798i	0.839647i
$233$	48.6597i	0.208840i	0.994533	+	0.104420i	$0.0332986\pi$
$233$	48.6597i	0.208840i	−0.994533	+	0.104420i	$0.966701\pi$
$234$	93.5946	0.399977
$235$	0	0
$236$	−94.5942	−0.400823
$237$	0	0
$238$	0	0
$239$	217.542	0.910219	0.455109	−	0.890436i	$-0.349600\pi$
$239$	217.542	0.910219	0.455109	+	0.890436i	$0.349600\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	72.7773i	0.300733i
$243$	− 251.453i	− 1.03479i
$244$	0	0
$245$	0	0
$246$	69.6321	0.283057
$247$	0	0
$248$	− 128.481i	− 0.518068i
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	−146.686	−0.577505
$255$	0	0
$256$	54.5736	0.213178
$257$	367.540i	1.43011i	0.699066	+	0.715057i	$0.253600\pi$
$257$	367.540i	1.43011i	−0.699066	+	0.715057i	$0.746400\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	280.578	1.07501
$262$	− 109.716i	− 0.418762i
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−303.541	−1.12840	−0.564202	−	0.825637i	$-0.690816\pi$
$269$	−303.541	−1.12840	−0.564202	+	0.825637i	$0.690816\pi$
$270$	0	0
$271$	−286.000	−1.05535	−0.527675	−	0.849446i	$-0.676936\pi$
$271$	−286.000	−1.05535	−0.527675	+	0.849446i	$0.676936\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	129.171	0.468010
$277$	− 549.049i	− 1.98213i	−0.133391	−	0.991063i	$-0.542587\pi$
$277$	− 549.049i	− 1.98213i	0.133391	−	0.991063i	$-0.457413\pi$
$278$	− 71.1559i	− 0.255956i
$279$	−185.058	−0.663290
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	− 87.1660i	− 0.309099i
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	512.687	1.80524
$285$	0	0
$286$	0	0
$287$	0	0
$288$	168.524i	0.585154i
$289$	−289.000	−1.00000
$290$	0	0
$291$	0	0
$292$	206.785i	0.708169i
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	−45.4938	−0.154741
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	540.872	1.80894
$300$	0	0
$301$	0	0
$302$	− 113.480i	− 0.375760i
$303$	256.244i	0.845689i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 214.000i	− 0.697068i	−0.937296	−	0.348534i	$-0.886680\pi$
$307$	− 214.000i	− 0.697068i	0.937296	−	0.348534i	$-0.113320\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	486.858	1.56546	0.782730	−	0.622361i	$-0.213826\pi$
$311$	486.858	1.56546	0.782730	+	0.622361i	$0.213826\pi$
$312$	− 166.769i	− 0.534517i
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	266.000i	0.839117i	0.907728	+	0.419558i	$0.137815\pi$
$317$	266.000i	0.839117i	−0.907728	+	0.419558i	$0.862185\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	81.2848	0.250879
$325$	0	0
$326$	−189.010	−0.579785
$327$	0	0
$328$	344.554i	1.05047i
$329$	0	0
$330$	0	0
$331$	661.999	2.00000	0.999999	−	0.00159277i	$-0.000506994\pi$
$331$	661.999	2.00000	0.999999	+	0.00159277i	$0.000506994\pi$
$332$	0	0
$333$	0	0
$334$	−145.555	−0.435792
$335$	0	0
$336$	0	0
$337$	0	0		−	1.00000i	$-0.5\pi$
$337$	0	0			1.00000i	$0.5\pi$
$338$	− 230.969i	− 0.683341i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	−13.2322	−0.0382435
$347$	602.000i	1.73487i	0.497550	+	0.867435i	$0.334233\pi$
$347$	602.000i	1.73487i	−0.497550	+	0.867435i	$0.665767\pi$
$348$	− 238.132i	− 0.684287i
$349$	26.0476	0.0746349	0.0373174	−	0.999303i	$-0.488119\pi$
$349$	26.0476	0.0746349	0.0373174	+	0.999303i	$0.488119\pi$
$350$	0	0
$351$	−566.911	−1.61513
$352$	0	0
$353$	695.320i	1.96974i	0.173284	+	0.984872i	$0.444562\pi$
$353$	695.320i	1.96974i	−0.173284	+	0.984872i	$0.555438\pi$
$354$	−24.1396	−0.0681908
$355$	0	0
$356$	0	0
$357$	0	0
$358$	− 172.733i	− 0.482494i
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	361.000	1.00000
$362$	0	0
$363$	− 186.780i	− 0.514546i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	271.164i	0.736859i
$369$	496.280	1.34493
$370$	0	0
$371$	0	0
$372$	157.062i	0.422210i
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	0	0
$375$	0	0
$376$	431.316	1.14712
$377$	− 997.120i	− 2.64488i
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	376.464	0.988095
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	186.814	0.486495
$385$	0	0
$386$	−21.7389	−0.0563184
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	− 225.113i	− 0.574267i
$393$	281.581i	0.716490i
$394$	131.998	0.335021
$395$	0	0
$396$	0	0
$397$	− 528.356i	− 1.33087i	−0.746455	−	0.665435i	$-0.768246\pi$
$397$	− 528.356i	− 1.33087i	0.746455	−	0.665435i	$-0.231754\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	657.660i	1.63191i
$404$	−603.948	−1.49492
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−256.398	−0.626889	−0.313445	−	0.949606i	$-0.601483\pi$
$409$	−256.398	−0.626889	−0.313445	+	0.949606i	$0.601483\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	−91.5402	−0.221112
$415$	0	0
$416$	598.902	1.43967
$417$	182.619i	0.437935i
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	− 244.195i	− 0.578661i
$423$	− 621.247i	− 1.46867i
$424$	0	0
$425$	0	0
$426$	130.833	0.307120
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	− 284.218i	− 0.657913i
$433$	0	0		−	1.00000i	$-0.5\pi$
$433$	0	0			1.00000i	$0.5\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	52.7697i	0.120479i
$439$	−626.871	−1.42795	−0.713976	−	0.700171i	$-0.753107\pi$
$439$	−626.871	−1.42795	−0.713976	+	0.700171i	$0.753107\pi$
$440$	0	0
$441$	−324.242	−0.735243
$442$	0	0
$443$	− 867.929i	− 1.95921i	−0.200938	−	0.979604i	$-0.564399\pi$
$443$	− 867.929i	− 1.95921i	0.200938	−	0.979604i	$-0.435601\pi$
$444$	0	0
$445$	0	0
$446$	−229.760	−0.515157
$447$	0	0
$448$	0	0
$449$	574.000	1.27840	0.639198	−	0.769042i	$-0.279266\pi$
$449$	574.000	1.27840	0.639198	+	0.769042i	$0.279266\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	291.241i	0.642916i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	542.680	1.17718	0.588590	−	0.808431i	$-0.299683\pi$
$461$	542.680	1.17718	0.588590	+	0.808431i	$0.299683\pi$
$462$	0	0
$463$	− 98.0000i	− 0.211663i	−0.994384	−	0.105832i	$-0.966250\pi$
$463$	− 98.0000i	− 0.211663i	0.994384	−	0.105832i	$-0.0337504\pi$
$464$	499.902	1.07738
$465$	0	0
$466$	−29.2671	−0.0628050
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	− 566.150i	− 1.20972i
$469$	0	0
$470$	0	0
$471$	0	0
$472$	− 119.448i	− 0.253067i
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	130.844i	0.273733i
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	440.227	0.909560
$485$	0	0
$486$	151.240	0.311194
$487$	− 238.236i	− 0.489190i	−0.969625	−	0.244595i	$-0.921345\pi$
$487$	− 238.236i	− 0.489190i	0.969625	−	0.244595i	$-0.0786550\pi$
$488$	0	0
$489$	485.086	0.991997
$490$	0	0
$491$	−301.732	−0.614526	−0.307263	−	0.951625i	$-0.599413\pi$
$491$	−301.732	−0.614526	−0.307263	+	0.951625i	$0.599413\pi$
$492$	− 421.201i	− 0.856101i
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−329.715	−0.664748
$497$	0	0
$498$	0	0
$499$	1.01465	0.00203337	0.00101668	−	0.999999i	$-0.499676\pi$
$499$	1.01465	0.00203337	0.00101668	+	0.999999i	$0.499676\pi$
$500$	0	0
$501$	373.560	0.745629
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	592.773i	1.16918i
$508$	887.298i	1.74665i
$509$	989.779	1.94456	0.972278	−	0.233827i	$-0.0751249\pi$
$509$	989.779	1.94456	0.972278	+	0.233827i	$0.0751249\pi$
$510$	0	0
$511$	0	0
$512$	516.912i	1.00959i
$513$	0	0
$514$	−221.062	−0.430082
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	33.9600	0.0654336
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	168.758i	0.323291i
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	−663.665	−1.26654
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−529.000	−1.00000
$530$	0	0
$531$	−172.047	−0.324005
$532$	0	0
$533$	− 1763.68i	− 3.30897i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	443.312i	0.825535i
$538$	− 182.569i	− 0.339348i
$539$	0	0
$540$	0	0
$541$	517.647	0.956834	0.478417	−	0.878133i	$-0.341211\pi$
$541$	517.647	0.956834	0.478417	+	0.878133i	$0.341211\pi$
$542$	− 172.019i	− 0.317379i
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 866.121i	− 1.58340i	−0.610909	−	0.791701i	$-0.709196\pi$
$547$	− 866.121i	− 1.58340i	0.610909	−	0.791701i	$-0.290804\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	163.109i	0.295487i
$553$	0	0
$554$	330.234	0.596091
$555$	0	0
$556$	−430.419	−0.774135
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	− 111.306i	− 0.199473i
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	−527.263	−0.934864
$565$	0	0
$566$	0	0
$567$	0	0
$568$	647.389i	1.13977i
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	210.698	0.365795
$577$	− 950.813i	− 1.64786i	−0.566694	−	0.823928i	$-0.691778\pi$
$577$	− 950.813i	− 1.64786i	0.566694	−	0.823928i	$-0.308222\pi$
$578$	− 173.824i	− 0.300733i
$579$	55.7920	0.0963593
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	−261.115	−0.447115
$585$	0	0
$586$	0	0
$587$	637.468i	1.08598i	0.839740	+	0.542988i	$0.182707\pi$
$587$	637.468i	1.08598i	−0.839740	+	0.542988i	$0.817293\pi$
$588$	275.190i	0.468010i
$589$	0	0
$590$	0	0
$591$	−338.768	−0.573212
$592$	0	0
$593$	286.000i	0.482293i	0.970489	+	0.241147i	$0.0775235\pi$
$593$	286.000i	0.482293i	−0.970489	+	0.241147i	$0.922476\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	325.316i	0.544007i
$599$	−1106.00	−1.84641	−0.923205	−	0.384307i	$-0.874440\pi$
$599$	−1106.00	−1.84641	−0.923205	+	0.384307i	$0.874440\pi$
$600$	0	0
$601$	−1201.97	−1.99995	−0.999973	−	0.00737547i	$-0.997652\pi$
$601$	−1201.97	−1.99995	−0.999973	+	0.00737547i	$0.997652\pi$
$602$	0	0
$603$	0	0
$604$	−686.434	−1.13648
$605$	0	0
$606$	−154.122	−0.254326
$607$	386.000i	0.635914i	0.948105	+	0.317957i	$0.102997\pi$
$607$	386.000i	0.635914i	−0.948105	+	0.317957i	$0.897003\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2207.79	−3.61341
$612$	0	0
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	128.714	0.209631
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	554.467	0.892862
$622$	292.829i	0.470785i
$623$	0	0
$624$	−427.973	−0.685854
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	626.717i	0.990074i
$634$	−159.990	−0.252350
$635$	0	0
$636$	0	0
$637$	1152.29i	1.80894i
$638$	0	0
$639$	932.469	1.45926
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 230.059i	− 0.355578i	−0.984069	−	0.177789i	$-0.943106\pi$
$647$	− 230.059i	− 0.355578i	0.984069	−	0.177789i	$-0.0568944\pi$
$648$	102.641i	0.158397i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	1143.31i	1.75355i
$653$	− 56.2240i	− 0.0861010i	−0.999073	−	0.0430505i	$-0.986292\pi$
$653$	− 56.2240i	− 0.0861010i	0.999073	−	0.0430505i	$-0.0137076\pi$
$654$	0	0
$655$	0	0
$656$	884.215	1.34789
$657$	376.098i	0.572448i
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	398.170i	0.601465i
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	975.233i	1.46212i
$668$	880.454i	1.31804i
$669$	589.669	0.881419
$670$	0	0
$671$	0	0
$672$	0	0
$673$	− 1323.09i	− 1.96596i	−0.183700	−	0.982982i	$-0.558807\pi$
$673$	− 1323.09i	− 1.96596i	0.183700	−	0.982982i	$-0.441193\pi$
$674$	0	0
$675$	0	0
$676$	−1397.12	−2.06675
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	788.189i	1.15401i	0.816741	+	0.577005i	$0.195779\pi$
$683$	788.189i	1.15401i	−0.816741	+	0.577005i	$0.804221\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	554.000	0.801737	0.400868	−	0.916136i	$-0.368708\pi$
$691$	554.000	0.801737	0.400868	+	0.916136i	$0.368708\pi$
$692$	80.0413i	0.115667i
$693$	0	0
$694$	−362.082	−0.521732
$695$	0	0
$696$	300.698	0.432037
$697$	0	0
$698$	15.6667i	0.0224451i
$699$	75.1129	0.107458
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	− 340.977i	− 0.485723i
$703$	0	0
$704$	0	0
$705$	0	0
$706$	−418.211	−0.592367
$707$	0	0
$708$	146.019i	0.206242i
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 643.224i	− 0.902138i
$714$	0	0
$715$	0	0
$716$	−1044.85	−1.45929
$717$	− 335.806i	− 0.468349i
$718$	0	0
$719$	862.000	1.19889	0.599444	−	0.800417i	$-0.295389\pi$
$719$	862.000	1.19889	0.599444	+	0.800417i	$0.295389\pi$
$720$	0	0
$721$	0	0
$722$	217.129i	0.300733i
$723$	0	0
$724$	0	0
$725$	0	0
$726$	112.342	0.154741
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	−187.076	−0.256620
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	0	0
$736$	−585.757	−0.795865
$737$	0	0
$738$	298.495i	0.404465i
$739$	−1316.84	−1.78192	−0.890958	−	0.454086i	$-0.849966\pi$
$739$	−1316.84	−1.78192	−0.890958	+	0.454086i	$0.849966\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	−198.328	−0.266570
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	− 1106.87i	− 1.47190i
$753$	0	0
$754$	599.734	0.795403
$755$	0	0
$756$	0	0
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1475.62	1.93906	0.969529	−	0.244977i	$-0.0787804\pi$
$761$	1475.62	1.93906	0.969529	+	0.244977i	$0.0787804\pi$
$762$	226.430i	0.297152i
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	611.421i	0.797159i
$768$	− 84.2418i	− 0.109690i
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	567.348	0.735860
$772$	131.498i	0.170334i
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	− 1022.18i	− 1.30547i
$784$	−577.697	−0.736859
$785$	0	0
$786$	−169.361	−0.215472
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	− 798.451i	− 1.01326i
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	317.788	0.400237
$795$	0	0
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	−395.560	−0.490769
$807$	468.557i	0.580615i
$808$	− 762.627i	− 0.943845i
$809$	−146.000	−0.180470	−0.0902349	−	0.995921i	$-0.528762\pi$
$809$	−146.000	−0.180470	−0.0902349	+	0.995921i	$0.528762\pi$
$810$	0	0
$811$	−1620.09	−1.99764	−0.998820	−	0.0485755i	$-0.984532\pi$
$811$	−1620.09	−1.99764	−0.998820	+	0.0485755i	$0.984532\pi$
$812$	0	0
$813$	441.480i	0.543026i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	− 154.214i	− 0.188526i
$819$	0	0
$820$	0	0
$821$	1274.00	1.55177	0.775883	−	0.630877i	$-0.217305\pi$
$821$	1274.00	1.55177	0.775883	+	0.630877i	$0.217305\pi$
$822$	0	0
$823$	− 437.797i	− 0.531952i	−0.963980	−	0.265976i	$-0.914306\pi$
$823$	− 437.797i	− 0.531952i	0.963980	−	0.265976i	$-0.0856942\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	553.723i	0.668747i
$829$	1654.00	1.99517	0.997587	−	0.0694210i	$-0.0221152\pi$
$829$	1654.00	1.99517	0.997587	+	0.0694210i	$0.0221152\pi$
$830$	0	0
$831$	−847.533	−1.01989
$832$	− 748.780i	− 0.899976i
$833$	0	0
$834$	−109.839	−0.131701
$835$	0	0
$836$	0	0
$837$	674.191i	0.805485i
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	956.883	1.13779
$842$	0	0
$843$	0	0
$844$	−1477.13	−1.75015
$845$	0	0
$846$	373.659	0.441677
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	− 791.403i	− 0.928877i
$853$	1606.00i	1.88277i	0.337339	+	0.941383i	$0.390473\pi$
$853$	1606.00i	1.88277i	−0.337339	+	0.941383i	$0.609527\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1589.30i	− 1.85449i	−0.374458	−	0.927244i	$-0.622171\pi$
$857$	− 1589.30i	− 1.85449i	0.374458	−	0.927244i	$-0.377829\pi$
$858$	0	0
$859$	845.930	0.984784	0.492392	−	0.870374i	$-0.336123\pi$
$859$	845.930	0.984784	0.492392	+	0.870374i	$0.336123\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1395.38i	1.61690i	0.588568	+	0.808448i	$0.299692\pi$
$863$	1395.38i	1.61690i	−0.588568	+	0.808448i	$0.700308\pi$
$864$	613.956	0.710597
$865$	0	0
$866$	0	0
$867$	446.111i	0.514546i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	319.201	0.364385
$877$	− 1558.00i	− 1.77651i	−0.459350	−	0.888255i	$-0.651918\pi$
$877$	− 1558.00i	− 1.77651i	0.459350	−	0.888255i	$-0.348082\pi$
$878$	− 377.041i	− 0.429432i
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	− 195.020i	− 0.221112i
$883$	− 938.000i	− 1.06229i	−0.847282	−	0.531144i	$-0.821762\pi$
$883$	− 938.000i	− 1.06229i	0.847282	−	0.531144i	$-0.178238\pi$
$884$	0	0
$885$	0	0
$886$	522.029	0.589198
$887$	841.479i	0.948680i	0.880342	+	0.474340i	$0.157313\pi$
$887$	841.479i	0.948680i	−0.880342	+	0.474340i	$0.842687\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	1389.81i	1.55808i
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 834.910i	− 0.930781i
$898$	345.241i	0.384456i
$899$	−1185.81	−1.31903
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	−175.171	−0.193346
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	0	0
$909$	−1098.45	−1.20842
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	−330.338	−0.358674
$922$	326.404i	0.354017i
$923$	− 3313.81i	− 3.59026i
$924$	0	0
$925$	0	0
$926$	58.9436	0.0636540
$927$	0	0
$928$	1079.87i	1.16365i
$929$	340.699	0.366737	0.183368	−	0.983044i	$-0.441300\pi$
$929$	340.699	0.366737	0.183368	+	0.983044i	$0.441300\pi$
$930$	0	0
$931$	0	0
$932$	177.036i	0.189952i
$933$	− 751.533i	− 0.805501i
$934$	0	0
$935$	0	0
$936$	714.898	0.763780
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	1724.97i	1.82924i
$944$	−306.533	−0.324717
$945$	0	0
$946$	0	0
$947$	− 1846.71i	− 1.95006i	−0.222069	−	0.975031i	$-0.571281\pi$
$947$	− 1846.71i	− 1.95006i	0.222069	−	0.975031i	$-0.428719\pi$
$948$	0	0
$949$	1336.58	1.40841
$950$	0	0
$951$	410.607	0.431764
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	791.471	0.827898
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−178.887	−0.186147
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1121.00i	1.15926i	0.814881	+	0.579629i	$0.196802\pi$
$967$	1121.00i	1.15926i	−0.814881	+	0.579629i	$0.803198\pi$
$968$	555.891i	0.574267i
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	− 914.846i	− 0.941200i
$973$	0	0
$974$	143.291	0.147116
$975$	0	0
$976$	0	0
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	291.763i	0.298326i
$979$	0	0
$980$	0	0
$981$	0	0
$982$	− 181.482i	− 0.184808i
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	531.867	0.540515
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	1154.00	1.16448	0.582240	−	0.813017i	$-0.302176\pi$
$991$	1154.00	1.16448	0.582240	+	0.813017i	$0.302176\pi$
$992$	− 712.236i	− 0.717980i
$993$	− 1021.89i	− 1.02909i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 1318.00i	− 1.32197i	−0.750401	−	0.660983i	$-0.770140\pi$
$997$	− 1318.00i	− 1.32197i	0.750401	−	0.660983i	$-0.229860\pi$
$998$	0.610277i	0 0.000611500i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	575.3.c.a.574.4		6
5.2	odd	4		575.3.d.b.551.2		3
5.3	odd	4		23.3.b.a.22.2	✓	3
5.4	even	2	inner	575.3.c.a.574.3		6
15.8	even	4		207.3.d.a.91.2		3
20.3	even	4		368.3.f.a.321.2		3
23.22	odd	2	CM	575.3.c.a.574.4		6
40.3	even	4		1472.3.f.b.321.2		3
40.13	odd	4		1472.3.f.a.321.2		3
115.22	even	4		575.3.d.b.551.2		3
115.68	even	4		23.3.b.a.22.2	✓	3
115.114	odd	2	inner	575.3.c.a.574.3		6
345.68	odd	4		207.3.d.a.91.2		3
460.183	odd	4		368.3.f.a.321.2		3
920.413	even	4		1472.3.f.a.321.2		3
920.643	odd	4		1472.3.f.b.321.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.3.b.a.22.2	✓	3	5.3	odd	4
23.3.b.a.22.2	✓	3	115.68	even	4
207.3.d.a.91.2		3	15.8	even	4
207.3.d.a.91.2		3	345.68	odd	4
368.3.f.a.321.2		3	20.3	even	4
368.3.f.a.321.2		3	460.183	odd	4
575.3.c.a.574.3		6	5.4	even	2	inner
575.3.c.a.574.3		6	115.114	odd	2	inner
575.3.c.a.574.4		6	1.1	even	1	trivial
575.3.c.a.574.4		6	23.22	odd	2	CM
575.3.d.b.551.2		3	5.2	odd	4
575.3.d.b.551.2		3	115.22	even	4
1472.3.f.a.321.2		3	40.13	odd	4
1472.3.f.a.321.2		3	920.413	even	4
1472.3.f.b.321.2		3	40.3	even	4
1472.3.f.b.321.2		3	920.643	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 575 = 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	575.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.601466i q^{2} -1.54364i q^{3} +3.63824 q^{4} +0.928445 q^{6} +4.59414i q^{8} +6.61718 q^{9} +O(q^{10})\)	\(q+0.601466i q^{2} -1.54364i q^{3} +3.63824 q^{4} +0.928445 q^{6} +4.59414i q^{8} +6.61718 q^{9} -5.61612i q^{12} -23.5162i q^{13} +11.7897 q^{16} +3.98001i q^{18} +23.0000i q^{23} +7.09168 q^{24} +14.1442 q^{26} -24.1073i q^{27} +42.4015 q^{29} -27.9663 q^{31} +25.4677i q^{32} +24.0749 q^{36} -36.3005 q^{39} +74.9986 q^{41} -13.8337 q^{46} -93.8839i q^{47} -18.1991i q^{48} -49.0000 q^{49} -85.5575i q^{52} +14.4997 q^{54} +25.5030i q^{58} -26.0000 q^{59} -16.8208i q^{62} +31.8410 q^{64} +35.5037 q^{69} +140.916 q^{71} +30.4003i q^{72} +56.8366i q^{73} -21.8335i q^{78} +22.3418 q^{81} +45.1091i q^{82} -65.4525i q^{87} +83.6795i q^{92} +43.1698i q^{93} +56.4679 q^{94} +39.3128 q^{96} -29.4718i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 24 q^{4} - 66 q^{6} - 54 q^{9}+O(q^{10}) \) 6 * q - 24 * q^4 - 66 * q^6 - 54 * q^9	\( 6 q - 24 q^{4} - 66 q^{6} - 54 q^{9} + 96 q^{16} + 264 q^{24} + 174 q^{26} + 510 q^{36} + 84 q^{39} - 294 q^{49} + 594 q^{54} - 156 q^{59} + 90 q^{64} + 486 q^{81} + 114 q^{94} - 42 q^{96}+O(q^{100}) \) 6 * q - 24 * q^4 - 66 * q^6 - 54 * q^9 + 96 * q^16 + 264 * q^24 + 174 * q^26 + 510 * q^36 + 84 * q^39 - 294 * q^49 + 594 * q^54 - 156 * q^59 + 90 * q^64 + 486 * q^81 + 114 * q^94 - 42 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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